For the graph of each function, find an equation of the line of symmetry and the coordinates of the vertex. Tell whether the value of the function at the vertex if a maximum or a minimum:
a) y = -x^2 + 4x - 7;
b) y = 2x^2 + 4x;
c) y = -3x^2 - 18x + 5.
For all these problems, we only have to complete the squares and transform the equations to the form
y=a(x-h)²+k
then (h,k) is the vertex.
If a>0, the vertex is a minimum, and
if a<0, then the vertex is a maximum.
a)
y=-x²+4x-7
=-1(x-2)²-3
So (h,k)=(2,-3) is the vertex.
Since a=-1, the vertex is a maximum.
I will leave (b) and (c) for you as exercises as they were intended.
To find the equation of the line of symmetry and coordinates of the vertex for each function, we need to use the vertex form of a quadratic equation, which is y = a(x - h)^2 + k. Here, (h, k) represents the coordinates of the vertex, and the line of symmetry is given by x = h. The value of a determines whether the vertex is a maximum or a minimum.
a) For the function y = -x^2 + 4x - 7:
To find the vertex, we can rewrite the function in the vertex form by completing the square.
y = -(x^2 - 4x) - 7
= -(x^2 - 4x + 4) - 7 + 4 [Adding and subtracting (4/2)^2 = 4]
Now, we can write it as:
y = -(x - 2)^2 - 3
From this equation, we can identify that the vertex is located at (2, -3). The line of symmetry is x = 2. Since the coefficient of x^2 is negative, the parabola opens downward, and the value of the function at the vertex (y = -3) is a maximum.
b) For the function y = 2x^2 + 4x:
To find the vertex, we can rewrite the function in the vertex form by completing the square.
y = 2(x^2 + 2x)
= 2(x^2 + 2x + 1) - 2 [Adding and subtracting (2/2)^2 = 1]
Now, we can write it as:
y = 2(x + 1)^2 - 2
From this equation, we can identify that the vertex is located at (-1, -2). The line of symmetry is x = -1. Since the coefficient of x^2 is positive, the parabola opens upward, and the value of the function at the vertex (y = -2) is a minimum.
c) For the function y = -3x^2 - 18x + 5:
To find the vertex, we can rewrite the function in the vertex form by completing the square.
y = -3(x^2 + 6x) + 5
= -3(x^2 + 6x + 9) + 5 + 27 [Adding and subtracting (6/2)^2 = 9]
Now, we can write it as:
y = -3(x + 3)^2 + 32
From this equation, we can identify that the vertex is located at (-3, 32). The line of symmetry is x = -3. Since the coefficient of x^2 is negative, the parabola opens downward, and the value of the function at the vertex (y = 32) is a maximum.
In summary:
a) Equation of line of symmetry: x = 2
Coordinates of the vertex: (2, -3)
The value of the function at the vertex is a maximum.
b) Equation of line of symmetry: x = -1
Coordinates of the vertex: (-1, -2)
The value of the function at the vertex is a minimum.
c) Equation of line of symmetry: x = -3
Coordinates of the vertex: (-3, 32)
The value of the function at the vertex is a maximum.